Representation Theory of Finite Groups

University Of California-Santa Cruz, Santa Cruz CA

Investigators

Abstract

The principal investigator is working in the field of representation theory of finite groups. Mainly he studies questions related to the conjectures of Alperin, Dade, and Broue. These conjectures state that certain invariants of a finite group and a fixed prime number can be determined by the same invariants of subgroups which are again related to that prime. The first conjectures are concerned with invariants that are integers, whereas the third conjecture predicts equivalences of categories. The first two conjectures and consequences of the third have been verified for a convincing number of examples. It is likely that these three conjectures are consequences of a single fact, object, theory, or construction which is still hidden. Probably more important than solving these particular conjectures would be the discovery of this hidden feature. The principal investigator works on general ideas of how to approach these conjectures. Graduate students are involved in this effort by writing computer programs which will either discard or give more evidence to the applicability of one of the suggested approaches. The principal investigator studies representations of groups. Groups can be viewed as mathematical abstractions of the notion of symmetry. They are among the most basic notions in many fields of mathematics and are applied widely, for example in cryptology, physics, chemistry, and computer science. Representations of groups are manifestations of a particular type of symmetry on other mathematical objects, as for example the four rotations and four reflections, that a square in the plane allows, form a representation of a group with 8 elements on a two-dimensional vector space. More specifically, the principal investigator studies mysterious coincidences in the representation theory of finite groups which have been discovered about fifteen years ago but could not be explained so far. This research is not aimed at immediate applications outside mathematics. However, in the history of the interplay between mathematics and other sciences, in particular with physics, it is a repeated pattern that theories and results which were considered as important within the edifice of mathematics became precisely what was needed in the other sciences in order to describe our real world. For example, group representations which have been studied a century ago became much later the right tool to describe particles in nuclear physics.